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Properties

Label 20.0.16902321876...0857.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 17^{15}$
Root discriminant $14.50$
Ramified primes $3, 17$
Class number $2$
Class group $[2]$
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 2, 3, -6, 23, -13, 7, 34, -50, 81, -50, 34, 7, -13, 23, -6, 3, 2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 2*x^18 + 3*x^17 - 6*x^16 + 23*x^15 - 13*x^14 + 7*x^13 + 34*x^12 - 50*x^11 + 81*x^10 - 50*x^9 + 34*x^8 + 7*x^7 - 13*x^6 + 23*x^5 - 6*x^4 + 3*x^3 + 2*x^2 - x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 2*x^18 + 3*x^17 - 6*x^16 + 23*x^15 - 13*x^14 + 7*x^13 + 34*x^12 - 50*x^11 + 81*x^10 - 50*x^9 + 34*x^8 + 7*x^7 - 13*x^6 + 23*x^5 - 6*x^4 + 3*x^3 + 2*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 2 x^{18} + 3 x^{17} - 6 x^{16} + 23 x^{15} - 13 x^{14} + 7 x^{13} + 34 x^{12} - 50 x^{11} + 81 x^{10} - 50 x^{9} + 34 x^{8} + 7 x^{7} - 13 x^{6} + 23 x^{5} - 6 x^{4} + 3 x^{3} + 2 x^{2} - x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(169023218768603112760857=3^{10}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.50$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{14} - \frac{1}{9} a^{12} - \frac{1}{3} a^{9} - \frac{4}{9} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{4}{9} a^{4} + \frac{2}{9} a^{2} + \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{27} a^{17} - \frac{1}{27} a^{16} + \frac{2}{27} a^{15} + \frac{4}{27} a^{14} + \frac{2}{27} a^{13} - \frac{2}{27} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{4}{27} a^{9} + \frac{13}{27} a^{8} + \frac{1}{3} a^{7} + \frac{11}{27} a^{5} - \frac{11}{27} a^{4} - \frac{13}{27} a^{3} + \frac{7}{27} a^{2} + \frac{1}{27} a - \frac{10}{27}$, $\frac{1}{27} a^{18} + \frac{1}{27} a^{16} - \frac{1}{9} a^{15} - \frac{1}{9} a^{14} - \frac{2}{27} a^{12} + \frac{5}{27} a^{10} - \frac{1}{3} a^{9} + \frac{4}{27} a^{8} + \frac{11}{27} a^{6} - \frac{1}{3} a^{5} - \frac{2}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{27} a^{2} - \frac{10}{27}$, $\frac{1}{81} a^{19} - \frac{1}{81} a^{17} - \frac{1}{81} a^{16} + \frac{2}{81} a^{15} + \frac{1}{81} a^{14} + \frac{1}{27} a^{13} + \frac{4}{81} a^{12} - \frac{13}{81} a^{11} + \frac{2}{9} a^{10} - \frac{8}{27} a^{9} - \frac{35}{81} a^{8} - \frac{34}{81} a^{7} + \frac{1}{3} a^{6} + \frac{8}{81} a^{5} - \frac{29}{81} a^{4} + \frac{7}{81} a^{3} - \frac{14}{81} a^{2} + \frac{5}{27} a - \frac{16}{81}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1314.30559263 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.44217.1, 5.1.44217.1 x5, 10.2.33237432513.1 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 5 sibling: 5.1.44217.1
Degree 10 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$17$17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$